In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.
Contents
Let (M, g) and (N, h) be two Riemannian manifolds and
a submersion.
Then f is a Riemannian submersion if and only if the isomorphism
is an isometry.
Examples
An example of a Riemannian submersion arises when a Lie group
Properties
The sectional curvature of the target space of a Riemannian submersion can be calculated from the curvature of the total space by O'Neill's formula:
where
In particular the lower bound for the sectional curvature of